Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #23984 |
| commutator, Frattini, cosocle | C23×D4 | C24 |
| prime radical | #346446 | C2 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
3126 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 3126 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10)(3,12,15,24)(4,9)(6,11,19,26)(7,17)(8,16,22,31)(13,21,14,30)(18,25)(20,32)(23,28,27,29);(2,7)(3,13)(4,17)(8,22)(9,10)(11,20)(12,24)(14,31)(15,28)(16,29)(18,32)(21,30)(23,27)(25,26);(4,18)(7,20)(9,25)(13,28)(14,29)(17,32)(21,27)(23,30);(1,3,6,15)(2,8,11,22)(4,14,18,29)(5,16,19,31)(7,21,20,27)(9,23,25,30)(10,24,26,12)(13,17,28,32)Polycyclic presentation:
2^10|a^2=f,d^2=i,e^2=j,b^a=be,c^b=cg,d^a=dfgh,d^b=dehij,e^a=eh,e^b=ej,e^c=ei,f^b=fhj,f^d=fij,f^e=fj,g^a=gi,g^d=gi,h^a=hj,h^d=hj58 conjugacy classes — view the full character table.
| maximal subgroup of | #72263157 |
| maximal normal subgroup of | #72263157 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);